47 Application to the solution of statically indeterminate systems

In a statically determinate structure the internal forces are determined uniquely by simple statical equilibrium considerations. This is not the case for a statically indeterminate system in which, as we have already noted, an infinite number of internal force or stress distributions may be found to satisfy the conditions of equilibrium. The true force system is, as we demonstrated in Section 4.5, the one satisfying the conditions of compatibility of displacement of the elastic structure or, alternatively, that for which the total complementary energy has a stationary value. We shall apply the principle to

a variety of statically indeterminate structures, beginning with the relatively simple singly-redundant pin-jointed frame shown in Fig. 4.12 in which each member has the same value of the product AE.

The first step is to choose the redundant member. In this example no advantage is gained by the choice of any particular member, although in some cases careful selection can result in a decrease in the amount of arithmetical labour. Taking BD as the redundant member we assume that it sustains a tensile force R due to the external loading. The total complementary energy of the framework is, with the notation of Eq. (4.15)

Hence

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